The status update pushing & pulling, profile, avatar and cover photo import are all controlled by the Social Profiles - JomSocial plugin. Disabling that isn't meant to be a long-term solution, but it does help narrow things down.
In your videos above, I'm not seeing any of the PHP warning or notice messages you mention. With error reporting set to 'maximum' and you try to authenticate with Twitter or Google, is there any error message shown on that blank white page? A PHP warning or notice really isn't a big deal. Those happen with some not quite perfect code, but don't 'break' things. Additionally, the messages you posted were all from JomSocial itself, and not JFBConnect.
What we're looking for is a PHP Fatal Error. That will 'break' the page and cause the white screen to appear. I'm not sure why none were shown in your videos, but all I can guess is that you didn't have error reporting set to 'maximum' at the time.
Can you try again and let us know what it is?
The next step you can try on your own is to go to the JFBConnect -> Profiles -> JomSocial area and set all of the field mappings to 'None' for Google+ and see if authentication starts working then. If not, try disabling all the other features (status pushing/pulling, avatar import, cover import, etc) until it starts working. Again, these are all steps to narrow down what feature is actually causing the 'break'.
I hope you can get an error message, but if not, the disabling of features step should really help figure out the cause. Again, I can't see this issue when I authenticate, so there's something special, it seems, going on with your Google and Twitter accounts. You may also want to try other social network accounts, just to try.
Thanks,
Alex
Multi Lingual Site - Trobules with Google+, Twitter
